Learning deformable 3D object models from single-view in-the-wild images has enabled impressive 3D shape reconstruction without supervision. However, it remains unclear whether these models capture the semantic structure required for downstream tasks. We find that existing deformable reconstruction approaches, despite producing visually plausible geometry, yield unstable correspondences across instances and perform poorly on semantic correspondence benchmarks. We introduce SEMAGIC, a framework for learning semantically consistent deformable 3D representations from single-view in-the-wild images. Rather than treating reconstruction as the end goal, SEMAGIC uses deformable modeling as a mechanism to discover category-level correspondences. Each category is represented by a canonical template mesh and a learned deformation field, functioning similarly to an autoencoder that reconstructs instance geometry from image features, enabling vertices to maintain consistent semantic meaning across instances. Semantic consistency is enforced during training through (i) a feature-level consistency loss aligning semantic features between canonical and deformed meshes, and (ii) vertex-index-conditioned deformation that preserves semantic correspondence across instances. By explicitly coupling geometric deformation with semantic alignment, SEMAGIC produces representations that maintain stable part correspondences across intra-category variation. Experiments demonstrate that SEMAGIC improves semantic correspondence of deformable models by +14.7 PCK@0.1 on SPair-71k, establishing deformable models as effective semantic 3D representations.